Sure, let's walk through the steps to calculate the tension in the string.
### Step 1: Identify the Given Information
- Mass of the stone, [tex]\( m = 0.1 \, \text{kg} \)[/tex]
- Radius of the circular path, [tex]\( r = 1.0 \, \text{m} \)[/tex]
- Rate of revolution, [tex]\( f = \frac{10}{\pi} \, \text{revolutions per second} \)[/tex]
### Step 2: Convert Revolutions per Second to Angular Velocity
To calculate the angular velocity ([tex]\( \omega \)[/tex]), we need to convert the rate of revolution to radians per second.
1 revolution is [tex]\( 2\pi \)[/tex] radians. Therefore, we convert the revolutions per second to radians per second using the formula:
[tex]\[ \omega = 2\pi \times f \][/tex]
[tex]\[
\omega = 2\pi \times \frac{10}{\pi} = 2\pi \times \frac{10}{\pi} = 20 \, \text{rad/s}
\][/tex]
### Step 3: Calculate the Centripetal Force (Tension)
The tension in the string that provides the centripetal force can be calculated using the formula:
[tex]\[ F = m \cdot \omega^2 \cdot r \][/tex]
Substitute the values into the equation:
[tex]\[
F = 0.1 \, \text{kg} \cdot (20 \, \text{rad/s})^2 \cdot 1.0 \, \text{m}
\][/tex]
[tex]\[
F = 0.1 \cdot 400 \cdot 1 = 40 \, \text{N}
\][/tex]
### Step 4: Conclusion
The tension in the string is [tex]\( 40 \, \text{N} \)[/tex]. This is the force required to keep the stone moving in a horizontal circular path at the given rate of revolution.
